NAME ______________________________________________ DATE______________ PERIOD _____
7-1
Study Guide and Intervention Multiplying Monomials
Multiply Monomials
A monomial is a number, a variable, or a product of a number and one or more variables. An expression of the form xn is called a power and represents the product you obtain when x is used as a factor n times. To multiply two powers that have the same base, add the exponents. For any number a and all integers m and n, am an a m n.
Product of Powers
Example 1
Example 2
Simplify (3x6)(5x2).
(3x6)(5x2) (3)(5)(x6 x2)
(4a3b)(3a2b5) (4)(3)(a3 a2)(b b5) 12(a3 2)(b1 5) 12a5b6
Group the coefficients and the variables
5)(x6 2)
(3 15x8
Simplify (⫺4a3b)(3a2b5).
Product of Powers Simplify.
The product is 12a5b6.
The product is 15x8.
Exercises Simplify. 2. n2 n7
3. (7x2)(x4)
4. x(x2)(x4)
5. m m5
6. (x3)(x4)
7. (2a2)(8a)
8. (rs)(rs3)(s2)
9. (x2y)(4xy3)
1 3
10. (2a3b)(6b3)
1 5
13. (5a2bc3) abc4
Chapter 7
11. (4x3)(5x7)
12. (3j 2k4)(2jk6)
14. (5xy)(4x2)( y4)
15. (10x3yz2)(2xy5z)
6
Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. y( y5)
NAME ______________________________________________ DATE______________ PERIOD _____
7-1
Study Guide and Intervention
(continued)
Multiplying Monomials Powers of Monomials
An expression of the form (xm) n is called a power of a power and represents the product you obtain when x m is used as a factor n times. To find the power of a power, multiply exponents. Power of a Power
For any number a and all integers m and n, (am) n ⫽ amn.
Power of a Product
For any number a and all integers m and n, (ab) m ⫽ amb m.
Example (⫺2ab2)3(a2)4 ⫽ ⫽ ⫽ ⫽ ⫽ The product is
(⫺2ab2)3(a8) (⫺2)3(a3)(b2)3(a8) (⫺2)3(a3)(a8)(b2)3 (⫺2)3(a11)(b2)3 ⫺8a11b6
Lesson 7-1
Simplify (⫺2ab2)3(a2)4. Power of a Power Power of a Product Group the coefficients and the variables Product of Powers Power of a Power
⫺8a11b6.
Exercises
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Simplify. 1. (y5) 2
2. (n7) 4
3. (x2) 5(x3)
4. ⫺3(ab4) 3
5. (⫺3ab4) 3
6. (4x2b) 3
7. (4a2)2(b3)
8. (4x) 2(b3)
9. (x2 y 4) 5
10. (2a3b2)(b3) 2
1 5
13. (25a2b) 3 ᎏ abc
2
16. (⫺2n6y5)(⫺6n3y2)(ny) 3
Chapter 7
11. (⫺4xy)3(⫺2x2)3
12. (⫺3j 2k3) 2(2j 2k) 3
14. (2xy)2(⫺3x2)(4y4)
15. (2x3y2z2)3(x2z)4
17. (⫺3a3n4)(⫺3a3n) 4
18. ⫺3(2x) 4(4x5y)2
7
Glencoe Algebra 1
NAME ______________________________________________ DATE______________ PERIOD _____
7-2
Study Guide and Intervention Dividing Monomials
Quotients of Monomials
To divide two powers with the same base, subtract the
exponents. am a
Quotient of Powers
m n. For all integers m and n and any nonzero number a, n a
Power of a Quotient
For any integer m and any real numbers a and b, b
a4b7 ab
Example 2
Simplify ᎏ . Assume 2
neither a nor b is equal to zero.
a4b7 a4 b7 a b2 ab2
(a4 1)(b7 2) a3b5 The quotient is a3b5 .
!
2a3b5 3 3b
. Simplify ᎏ 2
Assume that b is not equal to zero.
Group powers with the same base.
2a3b5 3 (2a3b5)3 3b2 (3b2)3
Quotient of Powers
Power of a Quotient
23(a3)3(b5)3 (3) (b )
Power of a Product
8a9b15 27b
Power of a Power
8a9b9 27
Quotient of Powers
3 2 3
Simplify.
6 8a9b9 27
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
The quotient is .
Exercises Simplify. Assume that no denominator is equal to zero. 3. 2
x5y3 x y
6. 5
2. 4
a2 a
5. 5 2
xy6 y x
8.
4.
2v5w3 4 v w
Chapter 7
2y7 14y
9. 2 2
r7s7t 2 s r t
3r 6s3 4 2r s
11. 5
4p4q4 3 3p q
2a2b 3 a
7. 4
10. 4 3
p5n4 p n
m6 m
55 5
1. 2
12. 3 3 2
13
Glencoe Algebra 1
Lesson 7-2
Example 1
a m am . b bm
0,
NAME ______________________________________________ DATE______________ PERIOD _____
7-2
Study Guide and Intervention
(continued)
Dividing Monomials Negative Exponents
Any nonzero number raised to the zero power is 1; for example, (⫺0.5)0 ⫽ 1. Any nonzero number raised to a negative power is equal to the reciprocal of the 1 number raised to the opposite power; for example, 6⫺3 ⫽ ᎏ3 . These definitions can be used 6 to simplify expressions that have negative exponents. Zero Exponent
For any nonzero number a, a0 ⫽ 1.
Negative Exponent Property
n For any nonzero number a and any integer n, a⫺n ⫽ ᎏ n and ᎏ ⫺n ⫽ a .
1 a
1
a
The simplified form of an expression containing negative exponents must contain only positive exponents. 4a⫺3b6 16a b c
Example
Simplify ᎏᎏ 2 6 ⫺5 . Assume that the denominator is not equal to zero.
1 4a⫺3b6 4 a⫺3 b6 ᎏ ᎏᎏ ⫽ ᎏ ᎏ ᎏ 2 6 ⫺5 16 16a b c a2 b6 c⫺5
Group powers with the same base.
1 4
Quotient of Powers and Negative Exponent Properties
1 4
Simplify.
⫽ ᎏ (a⫺3 ⫺ 2)(b6 ⫺ 6)(c5) ⫽ ᎏ a⫺5b0c5 1 1 4 a
⫽ ᎏ ᎏ5 (1)c5
Simplify.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
c5 4a
⫽ ᎏ5
Negative Exponent and Zero Exponent Properties
c5 4a
The solution is ᎏ5 .
Exercises Simplify. Assume that no denominator is equal to zero. 22 2
1. ᎏ ⫺3
m m
3. ᎏ 3
(⫺x⫺1 y)0 4w y
6. ᎏ ⫺2
(6a⫺1b)2 (b )
9. ᎏᎏ ⫺1 2 7
p⫺8 p
2. ᎏ ⫺4
4. ᎏ ⫺5
b⫺4 b
5. ᎏᎏ ⫺1 2
x4 y0 x
8. ᎏᎏ 2 4
7. ᎏ ⫺2 s⫺3t⫺5 (s t )
10. ᎏ 2 3 ⫺1
Chapter 7
(a2b3)2 (ab)
(3st)2u⫺4 s t u
4m2n2 0 8m ᐉ
11. ᎏ ⫺1
(⫺2mn2)⫺3 4m n
12. ᎏᎏ ⫺6 4
14
Glencoe Algebra 1
NAME ______________________________________________ DATE______________ PERIOD _____
7-3
Study Guide and Intervention Polynomials
Degree of a Polynomial
A polynomial is a monomial or a sum of monomials. A binomial is the sum of two monomials, and a trinomial is the sum of three monomials. Polynomials with more than three terms have no special name. The degree of a monomial is the sum of the exponents of all its variables. The degree of the polynomial is the same as the degree of the monomial term with the highest degree. Example
State whether each expression is a polynomial. If the expression is a polynomial, identify it as a monomial, binomial, or trinomial. Then give the degree of the polynomial. Expression
Polynomial?
Monomial, Binomial, or Trinomial?
Degree of the Polynomial
3x ⫺ 7xyz
Yes. 3x ⫺ 7xyz ⫽ 3x ⫹ (⫺7xyz), which is the sum of two monomials
binomial
3
Yes. ⫺25 is a real number.
monomial
0
3 n
none of these
—
trinomial
3
⫺25 7n3 ⫹ 3n⫺4
9x3 ⫹ 4x ⫹ x ⫹ 4 ⫹ 2x
No. 3n⫺4 ⫽ ᎏ4 , which is not a monomial Yes. The expression simplifies to 9x3 ⫹ 7x ⫹ 4, which is the sum of three monomials
Exercises
3 q
1. 36
2. ᎏ2 ⫹ 5
3. 7x ⫺ x ⫹ 5
4. 8g2h ⫺ 7gh ⫹ 2
1 4y
6. 6x ⫹ x2
5. ᎏ2 ⫹ 5y ⫺ 8 Find the degree of each polynomial. 7. 4x2y3z
8. ⫺2abc
9. 15m
10. s ⫹ 5t
11. 22
12. 18x2 ⫹ 4yz ⫺ 10y
13. x4 ⫺ 6x2 ⫺ 2x3 ⫺ 10
14. 2x3y2 ⫺ 4xy3
15. ⫺2r8s4 ⫹ 7r2s ⫺ 4r7s6
16. 9x2 ⫹ yz8
17. 8b ⫹ bc5
18. 4x4y ⫺ 8zx2 ⫹ 2x5
19. 4x2 ⫺ 1
20. 9abc ⫹ bc ⫺ d 5
21. h3m ⫹ 6h4m2 ⫺ 7
Chapter 7
20
Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
State whether each expression is a polynomial. If the expression is a polynomial, identify it as a monomial, binomial, or trinomial.
NAME ______________________________________________ DATE______________ PERIOD _____
7-3
Study Guide and Intervention
(continued)
Polynomials Write Polynomials in Order The terms of a polynomial are usually arranged so that the powers of one variable are in ascending (increasing) order or descending (decreasing) order. Example 1
Example 2
Arrange the terms of each polynomial so that the powers of x are in ascending order.
Arrange the terms of each polynomial so that the powers of x are in descending order.
a. x4 ⫺ x2 ⫹ 5x3 ⫺x2 ⫹ 5x3 ⫹ x4
a. x4 ⫹ 4x5 ⫺ x2 4x5 ⫹ x4 ⫺ x2
b. 8x3y ⫺ y2 ⫹ 6x2y ⫹ xy2 ⫺y2 ⫹ xy2 ⫹ 6x2y ⫹ 8x3y
b. ⫺6xy ⫹ y3 ⫺ x2y2 ⫹ x4y2 x4y2 ⫺ x2y2 ⫺ 6xy ⫹ y3
Exercises
1. 5x ⫹ x2 ⫹ 6
2. 6x ⫹ 9 ⫺ 4x2
3. 4xy ⫹ 2y ⫹ 6x2
4. 6y2x ⫺ 6x2y ⫹ 2
5. x4 ⫹ x3⫹ x2
6. 2x3 ⫺ x ⫹ 3x7
7. ⫺5cx ⫹ 10c2x3⫹ 15cx2
8. ⫺4nx ⫺ 5n3x3⫹ 5
9. 4xy ⫹ 2y ⫹ 5x2
Arrange the terms of each polynomial so that the powers of x are in descending order. 10. 2x ⫹ x2 ⫺ 5
11. 20x ⫺ 10x2 ⫹ 5x3
12. x2 ⫹ 4yx⫺ 10x5
13. 9bx ⫹ 3bx2 ⫺ 6x3
14. x3 ⫹ x5 ⫺ x2
15. ax2 ⫹ 8a2x5 ⫺ 4
16. 3x3y ⫺ 4xy2 ⫺ x4y2 ⫹ y5
17. x4 ⫹ 4x3 ⫺ 7x5 ⫹ 1
18. ⫺3x6 ⫺ x5 ⫹ 2x8
19. ⫺15cx2 ⫹ 8c2x5 ⫹ cx
20. 24x2y ⫺ 12x3y2 ⫹ 6x4
21. ⫺15x3 ⫹ 10x4y2 ⫹ 7xy2
Chapter 7
21
Glencoe Algebra 1
Lesson 7-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Arrange the terms of each polynomial so that the powers of x are in ascending order.
NAME ______________________________________________ DATE______________ PERIOD _____
7-4
Study Guide and Intervention Adding and Subtracting Polynomials
Add Polynomials
To add polynomials, you can group like terms horizontally or write them in column form, aligning like terms vertically. Like terms are monomial terms that are either identical or differ only in their coefficients, such as 3p and ⫺5p or 2x2y and 8x2y. Example 1
Example 2
Find (2x2 ⫹ x ⫺ 8) ⫹ (3x ⫺ 4x2 ⫹ 2).
Find (3x2 ⫹ 5xy) ⫹ (xy ⫹ 2x2).
Horizontal Method Group like terms.
Vertical Method Align like terms in columns and add.
(2x2 ⫹ x ⫺ 8) ⫹ (3x ⫺ 4x2 ⫹ 2) ⫽ [(2x2 ⫹ (⫺4x2)] ⫹ (x ⫹ 3x ) ⫹ [(⫺8) ⫹ 2)] ⫽ ⫺2x2 ⫹ 4x ⫺ 6.
3x2 ⫹ 5xy (⫹) 2x2 ⫹ xy
The sum is
⫺2x2
5x2
⫹ 4x ⫺ 6.
Put the terms in descending order.
⫹ 6xy
The sum is 5x2 ⫹ 6xy.
Exercises Find each sum. 2. (6x ⫹ 9) ⫹ (4x2 ⫺ 7)
3. (6xy ⫹ 2y ⫹ 6x) ⫹ (4xy ⫺ x)
4. (x2 ⫹ y2) ⫹ (⫺x2 ⫹ y2)
5. (3p2 ⫺ 2p ⫹ 3) ⫹ (p2 ⫺ 7p ⫹ 7)
6. (2x2 ⫹ 5xy ⫹ 4y2) ⫹ (⫺xy ⫺ 6x2 ⫹ 2y2)
7. (5p ⫹ 2q) ⫹ (2p2 ⫺ 8q ⫹ 1)
8. (4x2 ⫺ x ⫹ 4) ⫹ (5x ⫹ 2x2 ⫹ 2)
9. (6x2 ⫹ 3x) ⫹ (x2 ⫺ 4x ⫺ 3)
10. (x2 ⫹ 2xy ⫹ y2) ⫹ (x2 ⫺ xy ⫺ 2y2)
11. (2a ⫺ 4b ⫺ c) ⫹ (⫺2a ⫺ b ⫺ 4c)
12. (6xy2 ⫹ 4xy) ⫹ (2xy ⫺ 10xy2 ⫹ y2)
13. (2p ⫺ 5q) ⫹ (3p ⫹ 6q) ⫹ (p ⫺ q)
14. (2x2 ⫺ 6) ⫹ (5x2 ⫹ 2) ⫹ (⫺x2 ⫺ 7)
15. (3z2 ⫹ 5z) ⫹ (z2 ⫹ 2z) ⫹ (z ⫺ 4)
16. (8x2 ⫹ 4x ⫹ 3y2 ⫹ y) ⫹ (6x2 ⫺ x ⫹ 4y)
Chapter 7
28
Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. (4a ⫺ 5) ⫹ (3a ⫹ 6)
NAME ______________________________________________ DATE______________ PERIOD _____
7-4
Study Guide and Intervention
(continued)
Adding and Subtracting Polynomials Subtract Polynomials
You can subtract a polynomial by adding its additive inverse. To find the additive inverse of a polynomial, replace each term with its additive inverse or opposite. Example
Find (3x2 ⫹ 2x ⫺ 6) ⫺ (2x ⫹ x2 ⫹ 3).
Horizontal Method
Vertical Method
Use additive inverses to rewrite as addition. Then group like terms.
Align like terms in columns and subtract by adding the additive inverse.
(3x2 ⫹ 2x ⫺ 6) ⫺ (2x ⫹ x2 ⫹ 3) ⫽ (3x2 ⫹ 2x ⫺ 6) ⫹ [(⫺2x)⫹ (⫺x2) ⫹ (⫺3)] ⫽ [3x2 ⫹ (⫺x2)] ⫹ [2x ⫹ (⫺2x)] ⫹ [⫺6 ⫹ (⫺3)] ⫽ 2x2 ⫹ (⫺9) ⫽ 2x2 ⫺ 9
3x2 ⫹ 2x ⫺ 6 (⫺) x2 ⫹ 2x ⫹ 3
The difference is 2x2 ⫺ 9.
3x2 ⫹ 2x ⫺ 6 (⫹) ⫺x2 ⫺ 2x ⫺ 3 2x2 ⫺9 The difference is 2x2 ⫺ 9.
Exercises
1. (3a ⫺ 5) ⫺ (5a ⫹ 1)
2. (9x ⫹ 2) ⫺ (⫺3x2 ⫺ 5)
3. (9xy ⫹ y ⫺ 2x) ⫺ (6xy ⫺ 2x)
4. (x2 ⫹ y2) ⫺ (⫺x2 ⫹ y2)
5. (6p2 ⫹ 4p ⫹ 5) ⫺ (2p2 ⫺ 5p ⫹ 1)
6. (6x2 ⫹ 5xy ⫺ 2y2) ⫺ (⫺xy ⫺ 2x2 ⫺ 4y2)
7. (8p ⫺ 5q) ⫺ (⫺6p2 ⫹ 6q ⫺ 3)
8. (8x2 ⫺ 4x ⫺ 3) ⫺ (⫺2x ⫺ x2 ⫹ 5)
9. (3x2 ⫺ 2x) ⫺ (3x2 ⫹ 5x ⫺ 1)
10. (4x2 ⫹ 6xy ⫹ 2y2) ⫺ (⫺x2 ⫹ 2xy ⫺ 5y2)
11. (2h ⫺ 6j ⫺ 2k) ⫺ (⫺7h ⫺ 5j ⫺ 4k)
12. (9xy2 ⫹ 5xy) ⫺ (⫺2xy ⫺ 8xy2)
13. (2a ⫺ 8b) ⫺ (⫺3a ⫹ 5b)
14. (2x2 ⫺ 8) ⫺ (⫺2x2 ⫺ 6)
15. (6z2 ⫹ 4z ⫹ 2) ⫺ (4z2 ⫹ z)
16. (6x2 ⫺ 5x ⫹ 1) ⫺ (⫺7x2 ⫺ 2x ⫹ 4)
Chapter 7
29
Glencoe Algebra 1
Lesson 7-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Find each difference.
NAME ______________________________________________ DATE______________ PERIOD _____
7-5
Study Guide and Intervention Multiplying a Polynomial by a Monomial
Product of Monomial and Polynomial The Distributive Property can be used to multiply a polynomial by a monomial. You can multiply horizontally or vertically. Sometimes multiplying results in like terms. The products can be simplified by combining like terms. Example 1
Example 2
Find ⫺3x2(4x2 ⫹ 6x ⫺ 8).
Horizontal Method 3x2(4x2 6x 8) 3x2(4x2) (3x2)(6x) (3x2)(8) 12x4 (18x3) (24x2) 12x4 18x3 24x2 Vertical Method 4x2 6x 8 ( ) 3x2
Simplify ⫺2(4x 2 ⫹ 5x) ⫺ x(x2 ⫹ 6x).
2(4x2 5x) x( x2 6x) 2(4x2) (2)(5x) (x)(x2) (x)(6x) 8x2 (10x) (x3) (6x2) (x3) [8x2 (6x2)] (10x) x3 14x2 10x
12x4 18x3 24x2 The product is 12x4 18x3 24x2. Exercises
1. x(5x x2)
2. x(4x2 3x 2)
3. 2xy(2y 4x2)
4. 2g( g2 2g 2)
5. 3x(x4 x3 x2)
6. 4x(2x3 2x 3)
7. 4cx(10 3x)
8. 3y(4x 6x3 2y)
9. 2x2y2(3xy 2y 5x)
Simplify. 10. x(3x 4) 5x
11. x(2x2 4x) 6x2
12. 6a(2a b) 2a(4a 5b)
13. 4r(2r2 3r 5) 6r(4r2 2r 8)
14. 4n(3n2 n 4) n(3 n)
15. 2b(b2 4b 8) 3b(3b2 9b 18)
16. 2z(4z2 3z 1) z(3z2 2z 1)
17. 2(4x2 2x) 3(6x2 4) 2x(x 1)
Chapter 7
35
Glencoe Algebra 1
Lesson 7-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Find each product.
NAME ______________________________________________ DATE______________ PERIOD _____
7-5
Study Guide and Intervention
(continued)
Multiplying a Polynomial by a Monomial Solve Equations with Polynomial Expressions
Many equations contain polynomials that must be added, subtracted, or multiplied before the equation can be solved. Example
Solve 4(n 2) 5n
4(n ⫺ 2) ⫹ 5n ⫽ 6(3 ⫺ n) ⫹ 19 4n ⫺ 8 ⫹ 5n ⫽ 18 ⫺ 6n ⫹ 19 9n ⫺ 8 ⫽ 37 ⫺ 6n 15n ⫺ 8 ⫽ 37 15n ⫽ 45 n⫽3
6(3 n) 19.
Original equation Distributive Property Combine like terms. Add 6n to both sides. Add 8 to both sides. Divide each side by 15.
The solution is 3.
Exercises Solve each equation. 2. 3(x ⫹ 5) ⫺ 6 ⫽ 18
3. 3x(x ⫺ 5) ⫺ 3x2 ⫽ ⫺30
4. 6(x2 ⫹ 2x) ⫽ 2(3x2 ⫹ 12)
5. 4(2p ⫹ 1) ⫺ 12p ⫽ 2(8p ⫹ 12)
6. 2(6x ⫹ 4) ⫹ 2 ⫽ 4(x ⫺ 4)
7. ⫺2(4y ⫺ 3) ⫺ 8y ⫹ 6 ⫽ 4( y ⫺ 2)
8. c(c ⫹ 2) ⫺ c(c ⫺ 6) ⫽ 10c ⫺ 12
9. 3(x2 ⫺ 2x) ⫽ 3x2 ⫹ 5x ⫺ 11
10. 2(4x ⫹ 3) ⫹ 2 ⫽ ⫺4(x ⫹ 1)
11. 3(2h ⫺ 6) ⫺ (2h ⫹ 1) ⫽ 9
12. 3( y ⫹ 5) ⫺ (4y ⫺ 8) ⫽ ⫺2y ⫹ 10
13. 3(2a ⫺ 6) ⫺ (⫺3a ⫺ 1) ⫽ 4a ⫺ 2
14. 5(2x2 ⫺ 1) ⫺ (10x2 ⫺ 6) ⫽ ⫺(x ⫹ 2)
15. 3(x ⫹ 2) ⫹ 2(x ⫹ 1) ⫽ ⫺5(x ⫺ 3)
16. 4(3p2 ⫹ 2p) ⫺ 12p2 ⫽ 2(8p ⫹ 6)
Chapter 7
36
Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. 2(a ⫺ 3) ⫽ 3(⫺2a ⫹ 6)
NAME ______________________________________________ DATE______________ PERIOD _____
7-6
Study Guide and Intervention Multiplying Polynomials
Multiply Binomials
To multiply two binomials, you can apply the Distributive Property twice. A useful way to keep track of terms in the product is to use the FOIL method as illustrated in Example 2. Example 1
Example 2
Find (x ⫹ 3)(x ⫺ 4).
Find (x ⫺ 2)(x ⫹ 5) using the FOIL method.
Horizontal Method (x 3)(x 4) x(x 4) 3(x 4) (x)(x) x(4) 3(x) 3(4) x2 4x 3x 12 x2 x 12
(x 2)(x 5) First
Inner
Last
(x)(x) (x)(5) (2)(x) (2)(5) x2 5x (2x) 10 x2 3x 10 The product is x2 3x 10.
Vertical Method ( )
Outer
x 3 x 4 4x 12
x2 3x x2 x 12 The product is x2 x 12.
Exercises
1. (x 2)(x 3)
2. (x 4)(x 1)
3. (x 6)(x 2)
4. (p 4)(p 2)
5. (y 5)(y 2)
6. (2x 1)(x 5)
7. (3n 4)(3n 4)
8. (8m 2)(8m 2)
9. (k 4)(5k 1)
10. (3x 1)(4x 3)
11. (x 8)(3x 1)
12. (5t 4)(2t 6)
13. (5m 3n)(4m 2n)
14. (a 3b)(2a 5b)
15. (8x 5)(8x 5)
16. (2n 4)(2n 5)
17. (4m 3)(5m 5)
18. (7g 4)(7g 4)
Chapter 7
42
Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Find each product.
NAME ______________________________________________ DATE______________ PERIOD _____
7-6
Study Guide and Intervention
(continued)
Multiply Polynomials
The Distributive Property can be used to multiply any
two polynomials. Example
Find (3x ⫹ 2)(2x2 ⫺ 4x ⫹ 5).
(3x ⫹ 2)(2x2 ⫺ 4x ⫹ 5) ⫽ 3x(2x2 ⫺ 4x ⫹ 5) ⫹ 2(2x2 ⫺ 4x ⫹ 5) ⫽ 6x3 ⫺ 12x2 ⫹ 15x ⫹ 4x2 ⫺ 8x ⫹ 10 ⫽ 6x3 ⫺ 8x2 ⫹ 7x ⫹ 10
Distributive Property Distributive Property Combine like terms.
The product is 6x3 ⫺ 8x2 ⫹ 7x ⫹ 10.
Exercises
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Find each product. 1. (x ⫹ 2)(x2 ⫺ 2x ⫹ 1)
2. (x ⫹ 3)(2x2 ⫹ x ⫺ 3)
3. (2x ⫺ 1)(x2 ⫺ x ⫹ 2)
4. (p ⫺ 3)(p2 ⫺ 4p ⫹ 2)
5. (3k ⫹ 2)(k2 ⫹ k ⫺ 4)
6. (2t ⫹ 1)(10t2 ⫺ 2t ⫺ 4)
7. (3n ⫺ 4)(n2 ⫹ 5n ⫺ 4)
8. (8x ⫺ 2)(3x2 ⫹ 2x ⫺ 1)
9. (2a ⫹ 4)(2a2 ⫺ 8a ⫹ 3)
10. (3x ⫺ 4)(2x2 ⫹ 3x ⫹ 3)
11. (n2 ⫹ 2n ⫺ 1)(n2 ⫹ n ⫹ 2)
12. (t2 ⫹ 4t ⫺ 1)(2t2 ⫺ t ⫺ 3)
13. (y2 ⫺ 5y ⫹ 3)(2y2 ⫹ 7y ⫺ 4)
14. (3b2 ⫺ 2b ⫹ 1)(2b2 ⫺ 3b ⫺ 4)
Chapter 7
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Glencoe Algebra 1
Lesson 7-6
Multiplying Polynomials
